A diffusive predator-prey system with prey refuge and predator cannibalism

Yuxuan Zhang; Xinmiao Rong; Jimin Zhang; Yuxuan Zhang; Xinmiao Rong; Jimin Zhang

doi:10.3934/mbe.2019070

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 3: 1445-1470. doi: 10.3934/mbe.2019070

Previous Article Next Article

Research article Special Issues

A diffusive predator-prey system with prey refuge and predator cannibalism

1.
School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang, 150080, P.R. China
2.
School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, P.R. China
3.
Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin, Heilongjiang, 150080, P.R. China

Received: 18 December 2018 Accepted: 21 January 2019 Published: 21 February 2019

This paper is devoted to exploring a diffusive predator-prey system with prey refuge and predator cannibalism. We investigate dynamics of this system, including dissipation and persistence, local and global stability of constant steady states, Turing instability, and nonexistence and existence of nonconstant steady state solutions. The influence of prey refuge and predator cannibalism on predator and prey biomass density is also considered by using a systematic sensitivity analysis. Our studies suggest that appropriate predator cannibalism has a positive effect on predator biomass density, and then high predator cannibalism may stabilize the predator-prey ecosystem and prevent the paradox of enrichment.
- predator-prey system,
- predator cannibalism,
- prey refuge,
- steady states,
- stability
Citation: Yuxuan Zhang, Xinmiao Rong, Jimin Zhang. A diffusive predator-prey system with prey refuge and predator cannibalism[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1445-1470. doi: 10.3934/mbe.2019070

Related Papers:

Abstract

This paper is devoted to exploring a diffusive predator-prey system with prey refuge and predator cannibalism. We investigate dynamics of this system, including dissipation and persistence, local and global stability of constant steady states, Turing instability, and nonexistence and existence of nonconstant steady state solutions. The influence of prey refuge and predator cannibalism on predator and prey biomass density is also considered by using a systematic sensitivity analysis. Our studies suggest that appropriate predator cannibalism has a positive effect on predator biomass density, and then high predator cannibalism may stabilize the predator-prey ecosystem and prevent the paradox of enrichment.

References

[1]	W. Chen and M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Model., 42 (2005), 31–44.
[2]	M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington- DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15–39.
[3]	S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of Michaelis-Menten type ratio-dependent predator-prey system, J. Math. Biol., 42 (2003), 489–506.
[4]	N. Min and M. Wang, Hopf bifurcation and steady-state bifurcation for a Leslie-Gower preypredator model with strong Allee effect in prey, Discrete Cont. Dyn-A., 39 (2018), 1071–1099.
[5]	W. Ni and M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differ. Equations, 261 (2016), 4244–4274.
[6]	X. Yan and C. Zhang, Stability and turing instability in a diffusive predator-prey system with Beddington-DeAngelis functional response, Nonlinear Analysis RWA, 20 (2014), 1–13.
[7]	F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944–1977.
[8]	J. Ohlberger, O. Langangen, N. C. Stenseth, L. A. Vollestad, Community-level consequences of cannibalism, Am. Nat., 180 (2012), 791–801.
[9]	V. H. W. Rudolf, M. Kamo and M. Boots, Cannibals in space: the coevolution of cannibalism and dispersal in spatially structured populations, Am. Nat., 175 (2010), 513–524.
[10]	A. Basheer, E. Quansah, S. Bhowmick, et al., Prey cannibalism alters the dynamics of Holling- Tanner-type predator-prey models Nonlinear Dyn., 85 (2016), 2549–2567.
[11]	L. V. Shevtsova and G. A. Zhdanova, V.A. Movchan, A.B. Primak, Experimental interrelationship between Dreissena and planktic invertribates, Hydrobiol. J., 11 (1986), 36–39.
[12]	S. Chakraborty and J. Chattopadhyay, Effect of cannibalism on a predator-prey system with nutritional value: a model based study, Dyn. Syst., 26 (2011), 13–22.
[13]	M. A. Elgar and B. J. Crespi, Cannibalism: Ecology and Evolution Among Diverse Taxa, Oxford University Press, New York, 1992.
[14]	G. A. Polis, The evolution and dynamics of intraspecific predation, Annu. Rev. Ecol. Syst., 12 (1981), 225-251.
[15]	S. Fasani and S. Rinaldi, Remarks on cannibalism and pattern formation in spatially extended prey-predator systems, Nonlinear Dyn., 67 (2012), 2543–2548.
[16]	M. Genkai-Kato and N. Yamamura, Profitability of prey determines the response of population abundances to enrichment, Proc. R. Soc. Lond. B., 267 (2000), 2397–2401.
[17]	M. Genkai-Kato, Nutritional value of algae: A critical control on the stability of Daphnia-algal systems, J. Plankton Res., 26 (2004), 711–717.
[18]	C. Kohlmeier and W. Ebenhöh, The stabilizing role of cannibalism in a predator-prey system, Bull. Math. Biol., 57 (1995), 401–411.
[19]	G. Q. Sun, G. Zhang, Z. Jin, et al., Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dyn., 58 (2009), 75–84.
[20]	J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57 (1995), 63–76.
[21]	C. B. Huffaker and C. E. Kennett, Experimental studies on predation: predation and cyclamen-mite populations on strawberries in California, Hilgardia, 26 (1956).
[22]	L. Chen, F. Chen and L. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Analysis RWA, 11 (2010), 246–252.
[23]	A. P. Maiti, B. Dubey and J. Tushar, A delayed prey-predator model with Crowley-Martin-type functional response including prey refuge, Math. Method. Appl. Sci., 40 (2017), 5792–5809.
[24]	J. P. Tripathi, S. Abbas and M. Thakur, Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge, Nonlinear Dyn., 80 (2015), 177–196.
[25]	F. Wei and Q. Fu, Hopf bifurcation and stability for predator-prey systems with Beddington- DeAngelis type functional response and stage structure for prey incorporating refuge, Appl. Math. Model., 40 (2016), 126–134.
[26]	Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differ. Equations, 229 (2006), 63–91.
[27]	X. Chang and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge, Math. Biosci. Eng., 10 (2013), 979–996.
[28]	R. Cui, J. Shi and B. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Differ. Equations, 256 (2014), 108–129.
[29]	Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differ. Equations, 246 (2009), 3932–3956.
[30]	L. N. Guin and P. K. Mandal, Effect of prey refuge on spatiotemporal dynamics of the reactiondi ffusion system, Comput. Math. Appl. 68 (2014), 1325–1340.
[31]	X. Guan, W. Wang and Y. Cai, Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis RWA, 12 (2011), 2385–2395.
[32]	X. He and S. Zheng, Protection zone in a diffusive predator-prey model with Beddington- DeAngelis functional response, J. Math. Biol., 75 (2017), 239–257.
[33]	Y. Cai, Z. Gui, X. Zhang, et al., W. Wang, Bifurcations and pattern formation in a predator-prey model, Int. J. Bifurcat. Chaos, 28 (2018), 1850140.
[34]	K. D. Prasad and B. Prasad, Biological pest control using cannibalistic predators and with provision of additional food: a theoretical study, Theor. Ecol., 11 (2018), 191–211.
[35]	A. M. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B, 237 (1952), 37–72.
[36]	C. G. Jäger, S. Diehl and M. Emans, Physical determinants of phytoplankton production, algal stoichiometry, and vertical nutrient fluxes, Am. Nat., 175 (2010), 91–104.
[37]	R. Han and B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear crossdi ffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Analysis RWA, 45 (2019), 822–853.
[38]	M. Scheffer and S. Rinaldi, Minimal models of top-down control of phytoplankton, Freshwater Biol., 45 (2000), 265–283.
[39]	W. Wang, X. Gao, Y. Cai, et al., Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin I., 355 (2018), 7226–7245.
[40]	Y. Lou and W. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differ. Equations, 154 (1999), 157–190.
[41]	C. Lin,W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1988), 1–27.
[42]	L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Science, New York, 1973.
[43]	R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471–1488.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)